Problem 137 - Prealgebra and Introductory Algebra

Question

a. Name two numbers that are 4 units from 2 on the number line. b. Name two numbers that are 5 units from 3 on the number line.

Step-by-Step Solution

Answer

a. The numbers 4 units from 2 are -2 and 6. b. The numbers 5 units from 3 are -2 and 8.

Step 1: Determine the numbers for part a

For the number 4 units from 2, add and subtract 4 from 2. The two numbers are \(2+4=6\) and \(2-4=-2\). Thus, -2 and 6 are 4 units from 2.

Step 2: Determine the numbers for part b

For the number 5 units from 3, add and subtract 5 from 3. The two numbers are \(3+5=8\) and \(3-5=-2\). Thus, -2 and 8 are 5 units from 3.

Key Concepts

Distance on a Number Line

A number line is a visual representation of numbers in a straight line, where each point on the line corresponds to a real number. Understanding distance on a number line helps in visualizing and solving math problems involving spacing between numbers.
To find the distance between two numbers, simply count the number of steps it takes to move from one number to the other, regardless of direction. This makes it easy to compute how far one number is from another.

• For example, when finding numbers that are 4 units away from 2, you have to consider both directions. You can move 4 steps to the right, reaching 6, or 4 steps to the left, reaching -2.
• Similarly, for numbers 5 units away from 3, moving 5 steps right brings you to 8, and moving 5 steps left lands you at -2.

This concept of distance applies not just to whole numbers, but to any points on the number line, making it versatile for various math problems.

Integer Addition and Subtraction

Addition and subtraction of integers on a number line involves straightforward steps where movement to the right indicates addition, and movement to the left indicates subtraction.

• To add a number, start at your initial point and move to the right. For example, adding 4 to 2 on the number line gives you 6. Hence, start at 2 and count four spaces to the right to land at 6.
• To subtract, start at your initial point and move to the left. Subtracting 4 from 2, for instance, means starting at 2 and counting four spaces to the left, arriving at -2.

Understanding integer operations through this visualization provides clarity and enhances problem-solving abilities, making it easier to handle operations mentally and with confidence.
Using this approach can particularly be helpful when dealing with negative numbers, ensuring that subtraction does not become confusing.

Problem Solving Steps

Problem-solving in math, especially involving number lines, benefits from a systematic approach. Breaking down the task into smaller, manageable parts helps in achieving accurate results.
For problems like finding numbers at a certain distance, follow these steps:

• Identify the starting point on the number line. This is your reference number.
• Determine the number of units to move. This tells you how far and in which direction to go from the starting point.
• Add the units to move in one direction (to the right) and record the number.
• Subtract the units to move in the opposite direction (to the left) and record the number.

By following these steps, you can confidently determine the numbers that satisfy the conditions of a problem, like finding specific distances from a given point on the number line.
Such systematic approaches not only improve precision in calculations but also enhance one’s ability to tackle more complex problems efficiently.